DOUBLE-WIEBE FUNCTION PARAMETER ESTIMATION USING ANALYTICAL

In many cases the rate of combustion changes with respect to the crank angle, and a single Wiebe function does not sufficiently represent the MFB profile. To better match the experimental MFB profile, a double-Wiebe function has been used particularly in cases that have a non-symmetric characteristic in the combustion profile. In this paper, the step-by-step estimation of the double-Wiebe parameter is studied and developed for cases with a limited number of data points to represent the MFB profile using an analytical solution. This analytical solution method is a simple, robust and straight forward method to compute the double-Wiebe parameters, thus enabling direct application in the one-dimensional engine simulation tool that normally has computational limitations.

ANALYTICAL SOLUTION OF DOUBLE-WIEBE FUNCTION PARAMETERS FOR BURN DURATIONS OF ETHANOL-GASOLINE BLENDS IN A SI ENGINE

OVER VARIABLE COMPRESSION RATIOS AND EGR LEVELS

Authors: Yeliana, J Worm, D J Michalek, J D Naber

([email protected], [email protected], [email protected], [email protected])

Affiliations: Department of Mechanical Engineering - Engineering Mechanics, College of Engineering, 1400 Townsend Drive, Houghton, MI USA 49931-1295, Phone 906.487.2551, Fax 906.487.2822

Corresponding Author: [email protected]

Abstract

The mass fraction burn (MFB) profile represents the fraction of mass burned in the combustion chamber as a function of crank angle. The MFB profile is the key characteristic in spark ignition engines linking combustion rates to an important indicator of efficiency, emissions and overall engine performance. The Wiebe function is a function form fits the characteristics S-curve that is used to represent the MFB profile as a function of crankshaft position and is widely used in engine simulation. In many cases as the rate of combustion changes with crank angle, a single Wiebe function does not sufficiently represent the MFB profile. To obtain better agreement with the experimental MFB profile, a double-Wiebe function can be used, particularly in cases that have a non-symmetric combustion profile.

In comparison to the single-Wiebe function that has two parameters, the double-Wiebe function has five parameters, including two sets of single-Wiebe parameters and a mixture parameter that defines the weight of each Wiebe function in the MFB profile. In this paper, the estimation of the double-Wiebe parameter is studied and developed for cases with a limited number of data points that represent the MFB profile using a unique analytical solution. This method provides a robust and computationally efficient method for application in 1D engine simulation tools.

Keyword: Double-Wiebe Function, Combustion Modeling, Burn Durations, Ethanol, Spark Ignition Engine

Introduction

A heat release analysis is performed to gain better understanding of the combustion process in a spark ignition (SI) engine. This analysis is conducted by examining the in-cylinder pressure trace that is typically measured with a piezo-electric pressure transducer in response to engine geometry, operating conditions and combustion process. The mass fraction burn (MFB) profile, a characteristic S-shaped curve, represents the cumulative heat release during the combustion process. The derivative of this curve represents the rate at which the mass burns, starts at zero and increases after the spark discharge to a maximum level approximately halfway through the burning process and then decreases to zero as the combustion process ends. The slope of the first half of this curve is not necessary the same as the slope of the second half, particularly under high dilution and knocking combustion conditions, which have a characteristic non-symmetric combustion rate.

Considering the importance of the MFB profile to the analysis and simulation of SI engines, it is critical to accurately model the MFB profile. An accurate MFB profiles enables modeling of the combustion process and effective simulation of the overall engine for design, calibration and optimization purposes.

Experimental Design

The experimental data used for this research was obtained using the Michigan Tech Port Fuel Injected Cooperative Fuel Research (CFR) engine with variable compression ratio of 8:1 to16:1 [46] and a single cylinder Direct Injection-SI Hydra engine with a modern combustion chamber including variable compression ratio of 11:1 to 18.5:1 and variable valve timing (VVT) to control the residual fraction in the combustion chamber [21]. Both of these engines have been used in the ethanol-gasoline blend research [21, 38, 46]. Each engine has an AVL GH12D piezoelectric pressure transducer installed with an AVL PH01 flame arrestor to sense the in-cylinder pressure. Cylinder pressure data acquisition was accomplished using a DSP ACAP system [63]. Three hundred consecutive engine

cycles were recorded for every test point. Other high speed and low speed data, including fuel and airflow rates and various other critical pressures and temperatures, were measured and acquired using a combination of National Instruments (NI) hardware and software. A fully electronic control system was developed with Mototron’s Motohawk rapid prototyping engine control development environment [21, 39]. Mototron’s Mototune was used as the calibration tool and engine control unit (ECU) interface. The calibration tool was also used to record engine control parameters including intake manifold pressure, throttle position, air flow rate, commanded spark timing, fuel injection pressure, commanded injection duration, equivalence ratio, intake and exhaust cam phasing, and exhaust gas recirculation (EGR) level.

Wiebe Function

The Wiebe function is widely used in internal combustion engine applications to describe the fraction of mass burned in the combustion chamber during the combustion process [7]. This function starts at zero, indicating the start of combustion, and tends exponentially to one, indicating the end of combustion. Due to its simplicity, this function is used instead of the complicated turbulent reacting flame front calculation to predict the rate of combustion [2, 3, 7], and is often used in SI engine modeling to describe the MFB as a function of engine position (crank-angle) during the cycle [3, 7, 27, 54, 55].

The Wiebe function is expressed as:











 



 







 





1

exp 1

m o

b a

x 



 (V-16)

Where:

a Efficiency parameter m Shape factor

xb Mass fraction burn

 Crank angle

o Start of combustion

 Combustion duration

Previously, several methods to determine the Wiebe function parameters have been developed by fitting the Wiebe function to the MFB profile using a combination of the least squares method and direct algebraic solution [56]. It was observed that the efficiency parameter, “a” is not an independent variable, but is directly related to the combustion duration () [27, 56]. In this work,  is introduced as an efficiency parameter to represent both the “a” and , thus simplifying the formulation of the Wiebe function.











 



 



 





1

exp 1

m o

xb





 (V-17)



 a^1m1 (V-18)

Considering the “m+1” as , the Wiebe function has a similar form to the Weibull formula, a cumulative distribution function that has two parameters that are widely used in reliability, failure and lifetime data analysis [64]. The Weibull distribution is expressed as:

 

^{1 exp t}

F t

 



    

     (V-19)

Where  is the scale parameter,  is the shape factor and  is the location parameter. The location parameter  is known (in this case, the spark timing that indicates the start of

combustion o in SI engine). Calculation of the Weibull parameters had been studied using linear and non-linear regressions [65], a graphical approach [66], and a weighted least squares method [67]. Wiebe function terms will be used toward the discussion in this paper.

A simple analytical solution of the Wiebe function parameters is obtained by rearranging terms and taking the natural log twice of Equation (V-17) twice:



m1

 

ln _o

 

 m1

  

ln ln



ln



1x_b

 

(V-20) Equation (V-20) shows that “m+1” is the slope of the plot of ln(-ln(1-xb)) versus ln(-o)

with the intercept on the y-axis and the x-axis is (-(m+1) ln()) and ln(). The analytical solution of “m” and  from the linearized-equation are:

 



¹2



1 2

ln ln 1

ln 1 1

ln

b b AS

o o

x m x

 

 

  

  

 

 

  

  

 

(V-21)



1 ln

 

1



ln



ln 1



1

 

exp 1

o b

AS

m x

m

  ^ ^  ^{    }

  

  (V-22)

Where subscript 1 and 2 refer to the data points 1 and 2, respectively.

Double-Wiebe Function

The double-Wiebe function has been used in HCCI [57, 58] and in knocking cases for SI on single-Wiebe function for these non-symmetric combustion profiles. The double-Wiebe function parameters are estimated by fitting the MFB to the double-Wiebe function. The least squares method has been developed to compute these parameters, which were obtained by finding the minimum root mean square error (RMSE) over a given set of data points that satisfied the double-Wiebe equation [68]. However, in this work, estimation of

the parameters for the double-Wiebe function is developed using an analytical solution given a limited number of data points from the MFB profile.

Several methods to combine more than one Weibull function have been proposed [61, 69-71]. The mixture Weibull distribution parameters were obtained using a graphical approach [60, 72]. These applications are widely used in mechanical failure and reliability analysis.

In this paper, the double-Wiebe function was combined resulting two efficiency (1 and

1) and two shape parameters (m1 and m2) and a weighting factor (p) as shown in Equation (V-23). This weighting factor reflects the influence of each Wiebe function on the overall MFB profile.

 

_^

Analytical Solution Approach

Two general types of combustion characteristics can be identified by observing five location of MFB (10%, 25%, 50%, 75%, and 90%). The first type has a slower burn in the first half of the combustion period as compared to the second half of the combustion period, while the second type has a faster burn in the first half as compared to the second half of the combustion period. Figure V-16a shows a profile of MFB observed in the CFR engine (ethanol content of 85 %volume (E85), compression ratio (CR) = 8, spark advance (SA) = 10 ^oCA before top dead center (BTDC), EGR=10, intake cam center line (ICCL) = 112 ^oCA before gas exchange top dead center (BGETDC) and exhaust cam center line (ECCL) = 87.5 after gas exchange top dead center (AGETDC), speed at 900 RPM and net indicated mean effective pressure (NMEP) of 330 kPa). This exhibits a slower burning rate in the first half of the combustion period. Figure V-16b shows a MFB profile that has a faster burning rate in the first half as compared to the second half of the

combustion period. This was observed in the Hydra engine (E85, CR = 11, SA = 42.5

oCA BTDC, EGR = 0, ICCL = 110 ^oCA BGETDC and ECCL = 88 ^oCA AGETDC, speed at 1300 RPM and NMEP = 330 kPa). The red-crosses in both figures represent the locations of 10%, 25%, 50%, 75%, and 90% of MFB. These cases will be used as examples to demonstrate the analytical solution of the double-Wiebe function parameters estimation.

(a) (b)

Figure V-16 Combustion characteristics of two different engines

Given these locations, which correspond to five different MFB percentages, the Wiebe parameters can be estimated using Equations (V-21) and (V-22). This method is similar to plotting the ln(-ln(1-xb)) versus ln(-o). Figure V-8a and Figure V-8b show the plots of ln(-ln(1-xb)) versus ln(-o) of the five locations of MFB profile taken from the CFR and Hydra engines, respectively. Projection to the y-axis represents the MFB, and projection to the x-axis represents the burn duration at given MFB location. The slope of the line between any two points represents the “m+1” of Wiebe parameters.

-10 -5 0 5 10 15 20 25 30

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1





Mass Fraction Burn

-10 -5 0 5 10 15 20 25 300

0.01 0.02 0.03 0.04 0.05 0.06

Rate of Mass Fraction Burn

Crank Angle

CFR; E85; CR=8; SA=10^oCA BTDC; EGR=10; ICCL=112^oCA AGETDC; ECCL=-87.5^oCA BGETDC

-50 0 50 100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1





Mass Fraction Burn

-50 0 50 1000

0.01 0.02 0.03 0.04

Rate of Mass Fraction Burn

Crank Angle

Hydra; E85; CR=11; SA=42.5^oCA BTDC; EGR=0; ICCL=110^oCA AGETDC; ECCL=-86^oCA BGETDC

(a) (b) Figure V-17 Plot of ln(-ln(1-xb)) versus ln(-o)

The double-Wiebe parameter estimation process begins by assuming that there are two separate populations among the five MFB locations. The locations of 10% and 25% MFB describe the first population, which represents the first half of combustion period, and the location of 75% and 90% MFB describe the second population, which represents the second half of the combustion period. Figure V-9a and Figure V-9b show the same log plot for the CFR and Hydra engines, respectively. The green line represents the slope of the first population which represent the “m1+1” of the first Wiebe function, and the red line represents the slope of the second population, which represent the “m2+1” of the second Wiebe function. It is clearly shown that the engines have different combustion characteristics and the single-Wiebe function will not accurately represent the combustion processes. Figure V-19a and Figure V-19b show the profiles of both Wiebe functions for CFR and Hydra engines, respectively.

1.8 2.2 2.6 3 3.4 3.8 4.2

-3 -2 -1 0 1

10% MFB 25% MFB

50% MFB 75% MFB

90% MFB

ln( - _o) ln(-ln(1-xb))

CFR; E85; CR=8; SA=10^oCA BTDC; EGR=10; ICCL=112^oCA AGETDC; ECCL=-87.5^oCA BGETDC

2.6 3 3.4 3.8 4.2 4.6 5

-3 -2 -1 0 1

10% MFB 25% MFB

50% MFB 75% MFB

90% MFB

ln( - _o) ln(-ln(1-xb))

Hydra; E85; CR=11; SA=42.5^oCA BTDC; EGR=0; ICCL=110^oCA AGETDC; ECCL=-86^oCA BGETDC

(a) (b) Figure V-18 Log plot of ln(-ln(1-xb)) versus ln(-o)

(a) (b)

Figure V-19 Profile of both Wiebe function

Projection to the y-axis of the intersection of these two lines provides a good initial value for “p” the mixture parameter for the double-Wiebe function. From Figure V-9a and Figure V-9b, the “p” mixture parameters of the CFR and Hydra engines are 0.50 and 0.52, respectively. The estimated “p” mixture parameter can be expressed analytically as:

1.8 2.2 2.6 3 3.4 3.8 4.2

CFR; E85; CR=8; SA=10^oCA BTDC; EGR=10; ICCL=112^oCA AGETDC; ECCL=-87.5^oCA BGETDC

1^st Slope

Hydra; E85; CR=11; SA=42.5^oCA BTDC; EGR=0; ICCL=110^oCA AGETDC; ECCL=-86^oCA BGETDC

1^st Slope

Mass Fraction Burn

CFR; E85; CR=8; SA=10^oCA BTDC; EGR=10; ICCL=112^oCA AGETDC; ECCL=-87.5^oCA BGETDC

10-90%-1 = 22.38 m1 = 1.91

10-90%-2 = 14.58 m2 = 3.59

1^st Wiebe Function (AS) 2^nd Wiebe Function (AS)

-50 0 50 100

Mass Fraction Burn

Hydra; E85; CR=11; SA=42.5^oCA BTDC; EGR=0; ICCL=110^oCA AGETDC; ECCL=-86^oCA BGETDC

10-90%-1 = 26.90 m1 = 4.66

10-90%-2 = 55.32 m2 = 1.65

1^st Wiebe Function (AS) 2^nd Wiebe Function (AS)

        

Using the estimated “p” mixture parameter given by equation (V-24), the double-Wiebe function can be simplified as:

 

^_

Given a set of the first Wiebe function parameters, the second set of Wiebe function parameters can be easily determined using the following equation:



Figure V-20a and Figure V-20b show the plot of ln(-ln(1-xb)) versus ln(-o). The black line shows the double-Wiebe slope, which is a combination of the slopes of the first Wiebe function (green line) and the slope of the second Wiebe function (red line). The black line passes through the blue-dots, which represent the 10%, 25%, 50%, 75%, and 90% MFB locations.

(a) (b) Figure V-20 Log plot of ln(-ln(1-xb)) versus ln(-o)

Figure V-21a and Figure V-21b show the reconstructed MFB profiles using the double-Wiebe function estimation overlaid on the calculated MFB profiles from the experimentally measured pressure trace. It is shown that the double-Wiebe function with the parameters listed on the Figure matches the experimental MFB profile.

(a) (b)

Figure V-21 Mass fraction burn profile

1.8 2.2 2.6 3 3.4 3.8 4.2

CFR; E85; CR=8; SA=10^oCA BTDC; EGR=10; ICCL=112^oCA AGETDC; ECCL=-87.5^oCA BGETDC

1^st Slope 2^nd Slope Double Wiebe Slope

2.6 3 3.4 3.8 4.2 4.6 5

Hydra; E85; CR=11; SA=42.5^oCA BTDC; EGR=0; ICCL=110^oCA AGETDC; ECCL=-86^oCA BGETDC

1^st Slope 2^nd Slope Double Wiebe Slope

-10 -5 0 5 10 15 20 25 30

Mass Fraction Burn

CFR; E85; CR=8; SA=10^oCA BTDC; EGR=10; ICCL=112^oCA AGETDC; ECCL=-87.5^oCA BGETDC

10-90%-1 = 17.63 m1 = 1.91

10-90%-2 = 11.54 m2 = 5.60 p = 0.54

Experiment MFB Profile 1^st Wiebe Function 2^nd Wiebe Function Double-Wiebe Function

Mass Fraction Burn

Hydra; E85; CR=11; SA=42.5^oCA BTDC; EGR=0; ICCL=110^oCA AGETDC; ECCL=-86^oCA BGETDC

10-90%-1 = 25.56 m1 = 4.66

10-90%-2 = 45.71 m2 = 3.22 p = 0.52

Experiment MFB Profile 1^st Wiebe Function 2^nd Wiebe Function Double-Wiebe Function

Figure V-22a and Figure V-22b show the rate of MFB calculated using the double-Wiebe function approximation with the rate of MFB calculated from the measured pressure trace.

(a) (b)

Figure V-22 Mass fraction burn rate profile

Validation of Analytical Solution for Double-Wiebe Function

Validation was performed using the fitted Wiebe function to estimate the heat release, and from that the cylinder pressure computes during combustion. For this purpose a single-zone pressure model was developed by inverting the single-zone MFB analysis, which is derived from the ideal gas equation and the energy balance [68]. A single-Wiebe function estimation using a least squares method is compared to the double-Wiebe function parameters estimation using the analytical solution described above.

Figure V-23a and Figure V-23b show the MFB profile using the single-Wiebe function and double-Wiebe function compared to the experimental MFB profile. In both cases the double-Wiebe function provides a better estimation than the single-Wiebe function. This results in a more accurate reconstructed pressure profile. Figure V-24a and Figure V-24b show the reconstructed pressure traces obtained using the single-Wiebe and

double--10 -5 0 5 10 15 20 25 30

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Crank Angle

Mass Fraction Burn Rate

CFR; E85; CR=8; SA=10^oCA BTDC; EGR=10; ICCL=112^oCA AGETDC; ECCL=-87.5^oCA BGETDC

Experiment MFB Profile 1^st Wiebe Function 2^nd Wiebe Function Double-Wiebe Function

-50 0 50 100

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045

Crank Angle

Mass Fraction Burn Rate

Hydra; E85; CR=11; SA=42.5^oCA BTDC; EGR=0; ICCL=110^oCA AGETDC; ECCL=-86^oCA BGETDC

Experiment MFB Profile 1^st Wiebe Function 2^nd Wiebe Function Double-Wiebe Function

Wiebe function along with the experimentally measured pressure trace. Table V-5 shows the metrics calculated from the reconstructed pressure traces using the single-Wiebe and double-Wiebe functions with respect to the experimentally measured pressure trace for both cases which is chosen from two different engines. It is shown that the double-Wiebe function, which uses a simple analytical solution to estimate its parameters gives better results than the single-Wiebe function, which has parameters estimated using the least squares method.

(a) (b)

Figure V-23 Mass fraction burn profile

(a) (b)

Figure V-24 Pressure trace

-10 -5 0 5 10 15 20 25 30

Mass Fraction Burn

CFR; E85; CR=8; SA=10^oCA BTDC; EGR=10; ICCL=112^oCA AGETDC; ECCL=-87.5^oCA BGETDC

Experimental Pressure Trace Single-Wiebe Function (LSM) Double-Wiebe Fucntion (AS)

-50 0 50 100

Mass Fraction Burn

Hydra; E85; CR=11; SA=42.5^oCA BTDC; EGR=0; ICCL=110^oCA AGETDC; ECCL=-86^oCA BGETDC

Experimental Pressure Trace Single-Wiebe Function (LSM) Double-Wiebe Fucntion (AS)

-10 -5 0 5 10 15 20 25 30

CFR; E85; CR=8; SA=10^oCA BTDC; EGR=10; ICCL=112^oCA AGETDC; ECCL=-87.5^oCA BGETDC

Experimental Pressure Trace Single-Wiebe Function (LSM) Double-Wiebe Fucntion (AS)

-50 0 50 100

Hydra; E85; CR=11; SA=42.5^oCA BTDC; EGR=0; ICCL=110^oCA AGETDC; ECCL=-86^oCA BGETDC

Experimental Pressure Trace Single-Wiebe Function (LSM) Double-Wiebe Fucntion (AS)

Table V-5 Evaluation metrics of the reconstructed pressure traces using single-Wiebe and double-Wiebe functions

CFR Engine Hydra Engine

Single-Wiebe Function

(LSM)

Double-Wiebe Function

(AS)

Single-Wiebe Function

(LSM)

Double-Wiebe Function

(AS)

RMSE (kPa) 35.7 8.4 46.9 26.5

 NMEP (kPa) -1.3 0 0 -0.4

 Max Pressure (kPa) -76.8 20 134.6 87.1

Summary and Conclusions

A step-by step analytical solution to compute the five double-Wiebe function parameters has been discussed. Using the single-zone pressure model that was developed and validated by the author, the double-Wiebe function is shown to be more accurate than the single-Wiebe function which uses the least squares method to estimate two of its parameters.

The analytical approach requires carefully selected initial values of the MFB to accurately reconstruct the pressure trace. By assuming that the locations of 10% and 25%

MFB to represent the first Wiebe function and the locations of 75% and 90% MFB to represent the second Wiebe function, the double-Wiebe function can be simplified to a single-Wiebe function formula, which can be solved using a simple analytical solution.

Acknowledgements

The authors would like to thank Craig Marriott, Matthew Wiles, Kenneth Patton, Audley Brown, and Uwe Dieter Grebe of GM Advanced Powertrain for their support, discussion and feedback on this project. Additionally the authors and Michigan Technological University would like to acknowledge and thank the Fulbright scholarship program and the State of Michigan, through Michigan Energy Efficiency Grants (MIEEG Case No.U13129) for the support of the graduate students involved in this research.

VI. COMBUSTION MODEL INTEGRATION

There are several ways to develop and integrate the combustion model in GT-Power. In this report, a user compound was developed to accommodate the parametric predictive combustion model, which contains calculation of the Wiebe function parameters as a function of engine geometry and operating conditions. An RLT-dependence was used to connect the predictive combustion compound with the multi-Wiebe combustion template in the main engine model. The RLT-dependence was chosen because there were no signal ports available in the multi-Wiebe combustion template at the time this parametric combustion compound was being developed. Even though this parametric combustion compound was built in GT-Suite V6 built-12, this parametric combustion compound was ready for the GT-Suite V7 which has open ports in the multi-Wiebe combustion template, thus enables the direct connection in and out the multi-Wiebe template. Details of the parametric combustion models, both single-Wiebe and double-Wiebe parametric combustion models and their comparison can be found in reference [73].

INTEGRATION OF PARAMETRIC COMBUSTION MODEL OF ETHANOL-GASOLINE BLENDS OVER VARIABLE COMPRESSION

RATIOS AND VARIABLE CAM TIMING IN A SPARK IGNITION

RATIOS AND VARIABLE CAM TIMING IN A SPARK IGNITION

In document Parametric combustion modeling for ethanol-gasoline fuelled spark ignition engines (Page 149-196)